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ISTANBUL — Euphoria swept the world after the election of President Obama, a symbol of hope and yearning for compromise after years of war and resentment toward his predecessor’s style and policies. Today, after an electoral rebuke at home, Obama is still popular among America’s traditional allies, but his star power among Muslims — a focus of his international outreach — is fading.

American unhappiness with Obama and the government, evident in the staggering blow to incumbents in midterm elections Tuesday, stems largely from concern about the weak US economy, suggesting there will be a basic continuity in US foreign policy.

Still, Obama departs this week on a 10-day, four-country trip to Asia, his longest foreign trip as president, and pundits will keep a close watch for any signs that his weaker position at home is recalibrating his approach abroad. The Afghan war, Mideast peace efforts, Iran’s nuclear activities, climate change, and the prospect of a currency war rank among global challenges — with trade and finance topping the agenda at summits of world leaders in South Korea and Japan this month.

Democrats lost the House of Representatives to resurgent Republicans and suffered setbacks in the Senate, an outcome that will make it harder for Obama, faced with a divided government, to push his policies. Some in China fear an escalation of conflict over trade issues if the president seeks to deflect tougher Republican criticism of economic recovery plans.

“It is easier to accuse China of making that mess,’’ said Xiong Zhiyong, a professor at China Foreign Affairs University who specializes in US-China relations.

There is uncertainty, too, in the Middle East, where some Israelis believe Obama will have less leverage over them because of his party’s electoral losses. Many Israelis mistrust Obama, pointing to pressure on them to renew a slowdown on settlement construction as evidence that the United States is favoring the Palestinian side in its role as mediator.

Obama’s legislative achievements, which include an economic stimulus bill and a landmark health care measure, failed to dissipate frustration fueled by a sense that he has lost the electrifying power to inspire that he displayed as a candidate.

Yet in many countries, his call for multilateralism, which runs parallel to a decline in American diplomatic and economic clout, remains a welcome departure from the era of President Bush, whose two-term presidency was largely defined by the war in Iraq and the divisive debates on which it hinged. Obama’s brand has done much to repair America’s tarnished image, some citizens believe, even if concrete results have been lacking.

“People around the world were expecting him to be God,’’ said Mehmet Onol, a 29-year-old manager at the Istanbul branch of a New York-based consulting firm. “The great expectations are what make his term seem to be a disappointment.’’

For many Muslims, American policies do not differ markedly from one president to the next, and represent the hegemonic designs of a Western superpower, or even a vendetta against Islam. Obama has presided over the drawdown of US forces in Iraq, a plan set in motion by Bush, but a troop surge in Afghanistan and obstacles to a deal between Israel and the Palestinians appear to have undermined outreach to Muslims, embodied in his “new beginning’’ speech in Cairo last year.

On his trip to Asia, Obama will visit India, where Bush was popular for ending a three-decade ban on civilian nuclear trade that was slapped on the country after its first atomic test in 1974. Analysts anticipated closer cooperation on defense purchases and technology, but little if any changes in the alliance as a result of the American election.

“Notwithstanding the extreme partisanship in the United States on political issues, on foreign policy there is still a fair amount of cohesion,’’ said Kanti Bajpai, professor of international politics at New Delhi’s Jawaharlal Nehru University.

In this month’s Asia tour, Obama will probably get a warm welcome in Indonesia, where he lived as a child. Obama is popular in Brazil, where half the population is black, and in him they continue to see inspiration; Latin America’s biggest nation has few blacks in top political posts.

In Europe, meanwhile, there is a sense of bewilderment at the divisive culture of US politics.

“People here still don’t quite understand American politics, the idea of political majorities, Congress, or the US Constitution,’’ said Steven Fielding, University of Nottingham. “Obama is seen as the president and all-powerful.’’